3.1 Definition and Examples · Linear Algebra Refresher

Chapter 3 stops asking 'what is the answer' and starts asking 'what kind of object is the answer set'. A vector space is any set where you can add and scale sensibly — which lets one theorem cover

R^{n}

, matrices, polynomials, and functions simultaneously.

Definition · Vector space

A set

V

with operations

+

and scalar multiplication is a vector space if it is closed under both (C1:

x + y \in V

; C2:

α x \in V

) and the following hold for all

x, y, z \in V

and scalars

α, β

1 $x + y = y + x$
2 $(x + y) + z = x + (y + z)$
3There is $0 \in V$ with $x + 0 = x$
4For each $x$ there is $- x$ with $x + (- x) = 0$
5 $α (x + y) = α x + α y$
6 $(α + β) x = α x + β x$
7 $(α β) x = α (β x)$
8 $1 \cdot x = x$

→Shortcut

You will almost never verify all eight on an exam. The questions that matter ask you to disprove membership — and that always comes down to closure or the presence of

0

. Look for a concrete counterexample first.

The examples you must recognise

Space	Elements	Dimension	Zero vector
$R^{n}$	Column $n$ -tuples	$n$	$(0, \dots, 0)^{T}$
$R^{m \times n}$	$m \times n$ matrices	$mn$	The zero matrix $O$
$C [a, b]$	Continuous functions on $[a, b]$	infinite	The function $f (x) \equiv 0$
$P_{n}$	Polynomials of degree $< n$	$n$	The zero polynomial

Note the convention:

P_{n}

has dimension

n

, with basis

{1, x, \dots, x^{n - 1}}

. Polynomials of degree exactly

k

do not form a vector space.

Some consequences that follow from the axioms alone: $0 \cdot x = 0$ ; $x + y = 0$ forces $y = - x$ ; and $(- 1) x = - x$ . Also, if $α v = 0$ then either $α = 0$ or $v = 0$ — a fact used constantly in independence arguments.

Worked example

A set that fails to be a vector space

0/3 steps

S = {(x_{1}, x_{2})^{T} : x_{1} \geq 0, x_{2} \geq 0}

(the first quadrant) a vector space under the usual operations?

Check yourself

T / F

Every vector space contains a zero vector.

Which of these is not a vector space under the usual operations?

3.1Definition and Examples

What you must be able to do

The examples you must recognise

A set that fails to be a vector space

Check yourself